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 A204270 a(n) = tau(n)*Pell(n), where tau(n) = A000005(n), the number of divisors of n. 23
 1, 4, 10, 36, 58, 280, 338, 1632, 2955, 9512, 11482, 83160, 66922, 323128, 780100, 2354160, 2273378, 16465260, 13250218, 95966568, 154455860, 372889432, 450117362, 4346717760, 3935214363, 12667263848, 30581480180, 110745336312, 89120964298 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Compare g.f. to the Lambert series identity: Sum_{n>=1} x^n/(1-x^n) = Sum_{n>=1} tau(n)*x^n. Related identities: (1) Sum_{n>=1} n^k*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_{k}(n)*Pell(n)*x^n for k>=0. (2) Sum_{n>=1} phi(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Pell(n)*x^n. (3) Sum_{n>=1} moebius(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = x. (4) Sum_{n>=1} lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Pell(n^2)*x^(n^2). LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 FORMULA G.f.: Sum_{n>=1} Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} tau(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers. EXAMPLE G.f.: A(x) = 1 + 4*x + 10*x^2 + 36*x^3 + 58*x^4 + 280*x^5 + 338*x^6 +... where A(x) = x/(1-2*x-x^2) + 2*x^2/(1-6*x^2+x^4) + 5*x^3/(1-14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) + 29*x^5/(1-82*x^5-x^10) + 70*x^6/(1-198*x^6+x^12) +...+ Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +... MATHEMATICA Table[DivisorSigma[0, n] Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Jan 05 2018 *) PROG (PARI) /* Subroutines used in PARI programs below: */ {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)} {A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)} (PARI) {a(n)=sigma(n, 0)*Pell(n)} (PARI) {a(n)=polcoeff(sum(m=1, n, Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)} CROSSREFS Cf. A203847, A204271, A204272, A204273, A204274, A204275, A000005 (tau), A002203, A000045. Sequence in context: A152916 A222506 A108596 * A287220 A220205 A196880 Adjacent sequences:  A204267 A204268 A204269 * A204271 A204272 A204273 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 14 2012 STATUS approved

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Last modified June 22 21:29 EDT 2021. Contains 345393 sequences. (Running on oeis4.)